{"id":28333,"date":"2026-10-21T07:23:34","date_gmt":"2026-10-21T07:23:34","guid":{"rendered":"https:\/\/blog.oratrics.com\/?p=28333"},"modified":"2026-10-21T08:35:42","modified_gmt":"2026-10-21T08:35:42","slug":"first-ten-odd-numbers","status":"publish","type":"post","link":"https:\/\/oratrics.com\/blogs\/uncategorized\/first-ten-odd-numbers\/","title":{"rendered":"Sum of the First 10 Odd Numbers: 3 Easy Methods with Visual Proof"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"28333\" class=\"elementor elementor-28333\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8c6d29f elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"8c6d29f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e99245a\" data-id=\"e99245a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-5a097d9\" data-id=\"5a097d9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d21a6de elementor-widget elementor-widget-heading\" data-id=\"d21a6de\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Sum of the First 10 Odd Numbers: 3 Easy Methods with Visual Proof\n<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-062fa81 elementor-widget elementor-widget-text-editor\" data-id=\"062fa81\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The sum of the <\/span><b>first ten odd numbers<\/b><span style=\"font-weight: 400;\"> is <\/span><b>100<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">The numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17 and 19. Added together, they make 100, which is 10 \u00d7 10, or 10\u00b2.<\/span><\/p><p><span style=\"font-weight: 400;\">The <\/span><b>arithmetic mean (average)<\/b><span style=\"font-weight: 400;\"> of the first ten odd numbers is <\/span><b>10<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">Most people stop at the answer. This guide also shows you why it works, so you can solve the same kind of question for the first 20, 50 or 100 odd numbers.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3fd86f6 elementor-align-center elementor-widget elementor-widget-button\" data-id=\"3fd86f6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/demo.oratrics.com\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Book a Free Demo Today<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a5fca14 elementor-widget elementor-widget-image\" data-id=\"a5fca14\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"647\" src=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-1024x647.webp\" class=\"attachment-large size-large wp-image-24739\" alt=\"Speech on value of time with a confident schoolboy, clock, and hourglass illustrating the importance of time management.\" srcset=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-1024x647.webp 1024w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-300x190.webp 300w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-768x485.webp 768w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value-1536x970.webp 1536w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Time-Matters_-A-Speech-on-Value.webp 1578w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a130b79\" data-id=\"a130b79\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8fc29f3 elementor-widget elementor-widget-html\" data-id=\"8fc29f3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t\t<link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\r\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Poppins:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\r\n\r\n<div class=\"toc-wrapper\">\r\n    \r\n    \r\n    <div class=\"custom-toc\" id=\"custom-toc\">\r\n        <div class=\"toc-title\">\r\n            \u2630 Table of Contents\r\n        <\/div>\r\n        <ul class=\"toc-list\" id=\"toc-list\"><\/ul>\r\n    <\/div>\r\n    \r\n<\/div>\r\n\r\n<style>\r\n\/* Parent Fixed Wrapper holding both components together *\/\r\n.toc-wrapper {\r\n    position: fixed;\r\n    top: 150px; \r\n    z-index: 1 !important;\r\n    right: 70px;\r\n    width: 100%;\r\n    max-width: 340px;\r\n    font-family: 'Poppins', sans-serif;\r\n}\r\n\r\n\/* Image Styling *\/\r\n.toc-sticky-image {\r\n    width: 100%;\r\n    height: auto;\r\n    border-radius: 22px; \r\n    margin-bottom: 15px; \r\n    box-shadow: 0 4px 12px rgba(0,0,0,0.05);\r\n    display: block;\r\n}\r\n\r\n\/* TOC Container *\/\r\n.custom-toc {\r\n    background: #fff;\r\n    border: 1px solid #dfe3ea;\r\n    border-radius: 22px;\r\n    padding: 0px 28px;\r\n    width: 100%;\r\n    max-height: 24rem; \r\n    overflow-y: auto;\r\n    box-shadow: 0 2px 8px rgba(0,0,0,0.03);\r\n}\r\n\r\n\/* Title *\/\r\n.custom-toc .toc-title {\r\n    font-size: 25px;\r\n    font-weight: 700;\r\n    color: #17233c;\r\n    position: sticky;\r\n    padding: 0.6rem 0rem;\r\n    background-color: white;\r\n    top: 0px;\r\n    margin-bottom: 12px;\r\n    display: flex;\r\n    align-items: center;\r\n    gap: 10px;\r\n    z-index: 2;\r\n}\r\n\r\n\/* List *\/\r\n.custom-toc .toc-list {\r\n    list-style: none;\r\n    margin: 0;\r\n    padding: 0;\r\n}\r\n\r\n\/* Links *\/\r\n.custom-toc .toc-list li {\r\n    margin-bottom: 22px;\r\n}\r\n\r\n.custom-toc .toc-list a {\r\n    text-decoration: none;\r\n    color: #6b7a99;\r\n    font-size: 14px;\r\n    font-weight: 600;\r\n    transition: 0.3s ease;\r\n    display: inline-block;\r\n}\r\n\r\n\/* Hover Color *\/\r\n.custom-toc .toc-list a:hover {\r\n    color: #8B0000;\r\n}\r\n\r\n\/* H2 Indent *\/\r\n.custom-toc .toc-list .toc-h2 {\r\n    padding-left: 18px;\r\n    font-size: 14px;\r\n}\r\n\r\n\/* Mobile Responsive Adjustments *\/\r\n@media(max-width:768px) {\r\n    .toc-wrapper {\r\n        position: relative;\r\n        top: 0;\r\n        z-index: 1;\r\n        right: 0;\r\n        max-width: 100%;\r\n        margin-bottom: 30px;\r\n    }\r\n\r\n    .custom-toc {\r\n        max-height: none; \r\n        overflow-y: visible;\r\n    }\r\n\r\n    .custom-toc .toc-title {\r\n        font-size: 25px;\r\n    }\r\n\r\n    .custom-toc .toc-list a {\r\n        font-size: 14px;\r\n    }\r\n}\r\n<\/style>\r\n\r\n<script>\r\ndocument.addEventListener(\"DOMContentLoaded\", function () {\r\n    const tocList = document.getElementById(\"toc-list\");\r\n    const h1 = document.querySelector(\"h1\");\r\n    const h2s = document.querySelectorAll(\"h2\");\r\n\r\n    function createId(text) {\r\n        return text.toLowerCase().replace(\/[^a-z0-9]+\/g, '-');\r\n    }\r\n\r\n    if (h1) {\r\n        if (!h1.id) {\r\n            h1.id = createId(h1.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.innerHTML = `<a href=\"#${h1.id}\">${h1.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    }\r\n\r\n    h2s.forEach(h2 => {\r\n        if (!h2.id) {\r\n            h2.id = createId(h2.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.classList.add(\"toc-h2\");\r\n        li.innerHTML = `<a href=\"#${h2.id}\">${h2.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    });\r\n});\r\n<\/script>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6df059d elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"6df059d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-789b654\" data-id=\"789b654\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-310f24e\" data-id=\"310f24e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dbf1a89 elementor-widget elementor-widget-heading\" data-id=\"dbf1a89\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">What Are the First Ten Odd Numbers?<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5d488b9 elementor-widget elementor-widget-text-editor\" data-id=\"5d488b9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">An odd number is a whole number that can\u2019t be divided evenly by 2. It always leaves a remainder of 1, and it always ends in 1, 3, 5, 7 or 9.<\/span><\/p><p><span style=\"font-weight: 400;\">Counting up from 1, the first ten odd numbers are:<\/span><\/p><p><b>1, 3, 5, 7, 9, 11, 13, 15, 17, 19<\/b><\/p><p><span style=\"font-weight: 400;\">Each number is exactly 2 more than the one before it. That steady gap is what makes this list an arithmetic sequence, and it\u2019s the reason a neat shortcut exists.<\/span><\/p><p><span style=\"font-weight: 400;\">Odd numbers are easy to confuse with other number groups, so here\u2019s the difference:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Odd natural numbers<\/b><span style=\"font-weight: 400;\"> are the same list: 1, 3, 5, 7 and so on.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Consecutive odd numbers<\/b><span style=\"font-weight: 400;\"> are odd numbers with no odd number skipped between them.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Even numbers<\/b><span style=\"font-weight: 400;\"> (2, 4, 6, 8\u2026) belong to a different list entirely.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">If you\u2019re counting \u201cthe first ten odd numbers,\u201d you always start at 1 and stop at 19. The 10th odd number is 19, and the general rule for any position is <\/span><b>2n \u2212 1<\/b><span style=\"font-weight: 400;\">. Put n = 10 into that rule and you get 2(10) \u2212 1 = 19.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-d10ea8e\" data-id=\"d10ea8e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b60c8be elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"b60c8be\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-7b9c731\" data-id=\"7b9c731\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c204f35\" data-id=\"c204f35\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-bd9bc02 elementor-widget elementor-widget-heading\" data-id=\"bd9bc02\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Method 1: Add Them in Pairs<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-65c083c elementor-widget elementor-widget-text-editor\" data-id=\"65c083c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">You can add the numbers one by one: 1 + 3 = 4, then 4 + 5 = 9, and so on. It works, but it\u2019s slow and easy to slip on.<\/span><\/p><p><span style=\"font-weight: 400;\">Pairing is faster. Match the first number with the last, the second with the second-last, and so on:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">1 + 19 = 20<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">3 + 17 = 20<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">5 + 15 = 20<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">7 + 13 = 20<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">9 + 11 = 20<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">That gives you <\/span><b>five pairs, and every pair makes 20<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">5 \u00d7 20 = <\/span><b>100<\/b><\/p><p><span style=\"font-weight: 400;\">This is the same idea the young Carl Friedrich Gauss is famously said to have used to add up the numbers from 1 to 100. It works whenever the numbers rise by the same step each time.<\/span><\/p><p><b>Best for:<\/b><span style=\"font-weight: 400;\"> quick mental maths, and showing younger students that patterns beat brute force.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-98892e4\" data-id=\"98892e4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9c464cf elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"9c464cf\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-34fe751\" data-id=\"34fe751\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-acf012f\" data-id=\"acf012f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-06efe74 elementor-widget elementor-widget-heading\" data-id=\"06efe74\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Method 2: Use the Formula (The n\u00b2 Shortcut)<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d8bd1e8 elementor-widget elementor-widget-text-editor\" data-id=\"d8bd1e8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Here is the rule worth remembering:<\/span><\/p><p><b>The sum of the first n odd numbers = n\u00b2<\/b><\/p><p><span style=\"font-weight: 400;\">For the first ten odd numbers, n = 10:<\/span><\/p><p><span style=\"font-weight: 400;\">10\u00b2 = 10 \u00d7 10 = <\/span><b>100<\/b><\/p><p><span style=\"font-weight: 400;\">It works for any count. A few more examples:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">First 5 odd numbers: 5\u00b2 = 25 (1 + 3 + 5 + 7 + 9 = 25)<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">First 20 odd numbers: 20\u00b2 = 400<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">First 50 odd numbers: 50\u00b2 = 2,500<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">First 100 odd numbers: 100\u00b2 = 10,000<\/span><\/li><\/ul><h4><b>Why does the formula work?<\/b><\/h4><p><span style=\"font-weight: 400;\">You can prove it with the standard formula for the sum of an arithmetic progression:<\/span><\/p><p><span style=\"font-weight: 400;\">Sum = (number of terms \u00f7 2) \u00d7 (first term + last term)<\/span><\/p><p><span style=\"font-weight: 400;\">For the first n odd numbers, the first term is 1 and the last term is 2n \u2212 1. So:<\/span><\/p><p><span style=\"font-weight: 400;\">Sum = (n \u00f7 2) \u00d7 (1 + 2n \u2212 1) = (n \u00f7 2) \u00d7 2n = <\/span><b>n\u00b2<\/b><\/p><p><span style=\"font-weight: 400;\">Checking with n = 10: (10 \u00f7 2) \u00d7 (1 + 19) = 5 \u00d7 20 = 100. It matches Method 1 exactly. Pairing is the same calculation in disguise.<\/span><\/p><p><b>Best for:<\/b><span style=\"font-weight: 400;\"> exams, homework and any question that asks for a larger number of odd numbers.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-372ed51\" data-id=\"372ed51\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f4a1302 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"f4a1302\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-22fef51\" data-id=\"22fef51\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-2393d95\" data-id=\"2393d95\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6aef4d6 elementor-widget elementor-widget-heading\" data-id=\"6aef4d6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Method 3: See It with a Visual Proof<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b698039 elementor-widget elementor-widget-text-editor\" data-id=\"b698039\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Many learners understand this best when they see it. The picture explains why the answer is always a perfect square.<\/span><\/p><h4><b>Building a square, one layer at a time<\/b><\/h4><p><span style=\"font-weight: 400;\">Start with one square tile. That\u2019s your first odd number, <\/span><b>1<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">Now add an L-shaped border of <\/span><b>3<\/b><span style=\"font-weight: 400;\"> tiles around it. You get a 2 \u00d7 2 square, which has 4 tiles.<\/span><\/p><p><span style=\"font-weight: 400;\">Add an L-shaped border of <\/span><b>5<\/b><span style=\"font-weight: 400;\"> tiles. You get a 3 \u00d7 3 square, which has 9 tiles.<\/span><\/p><p><span style=\"font-weight: 400;\">Each new L-shape is the next odd number, and each time the shape stays a perfect square. Here is a 4 \u00d7 4 square, with each tile labeled by the layer it belongs to:<\/span><\/p><p><span style=\"font-weight: 400;\">1\u00a0 2\u00a0 3\u00a0 4<\/span><\/p><p><span style=\"font-weight: 400;\">2\u00a0 2\u00a0 3\u00a0 4<\/span><\/p><p><span style=\"font-weight: 400;\">3\u00a0 3\u00a0 3\u00a0 4<\/span><\/p><p><span style=\"font-weight: 400;\">4\u00a0 4\u00a0 4\u00a0 4<\/span><\/p><p><span style=\"font-weight: 400;\">Count the tiles in each layer:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Layer 1 has 1 tile<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Layer 2 has 3 tiles<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Layer 3 has 5 tiles<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Layer 4 has 7 tiles<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">Total: 1 + 3 + 5 + 7 = 16 = 4 \u00d7 4.<\/span><\/p><p><span style=\"font-weight: 400;\">Keep going until you\u2019ve added ten layers. The last L-shape has 19 tiles, and the finished shape is a <\/span><b>10 \u00d7 10 square with 100 tiles<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">So the picture makes the pattern clear: adding odd numbers in order always builds the next square number.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1bdd6cf\" data-id=\"1bdd6cf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f81e978 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"f81e978\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-6bda601\" data-id=\"6bda601\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-bec6746\" data-id=\"bec6746\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5f6976b elementor-widget elementor-widget-heading\" data-id=\"5f6976b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">The running total table\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-95b6060 elementor-widget elementor-widget-text-editor\" data-id=\"95b6060\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This table shows the \u201csquare number\u201d pattern building up:<\/span><\/p><table><tbody><tr><td><p><b>Odd numbers added<\/b><\/p><\/td><td><p><b>Number added<\/b><\/p><\/td><td><p><b>Running total<\/b><\/p><\/td><td><p><b>Same as<\/b><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">First 1<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">1<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">1<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">1\u00b2<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">First 2<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">3<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">4<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">2\u00b2<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">First 3<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">5<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">9<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">3\u00b2<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">First 4<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">7<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">16<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">4\u00b2<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">First 5<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">9<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">25<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">5\u00b2<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">First 6<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">11<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">36<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">6\u00b2<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">First 7<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">13<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">49<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">7\u00b2<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">First 8<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">15<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">64<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">8\u00b2<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">First 9<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">17<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">81<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">9\u00b2<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">First 10<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">19<\/span><\/p><\/td><td><p><b>100<\/b><\/p><\/td><td><p><b>10\u00b2<\/b><\/p><\/td><\/tr><\/tbody><\/table><p><span style=\"font-weight: 400;\">Every total in the third column is a perfect square. You can read the whole proof straight off the table.<\/span><\/p><p><b>Best for:<\/b><span style=\"font-weight: 400;\"> classroom teaching, visual learners, and anyone who wants to remember the rule instead of just using it.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e1f40b7\" data-id=\"e1f40b7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e182035 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"e182035\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-2a72fbb\" data-id=\"2a72fbb\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3914e31\" data-id=\"3914e31\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2acabaf elementor-widget elementor-widget-heading\" data-id=\"2acabaf\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Finding the Average of the First Ten Odd Numbers<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-01c5164 elementor-widget elementor-widget-text-editor\" data-id=\"01c5164\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Questions about the first ten odd numbers often ask for the <\/span><b>mean<\/b><span style=\"font-weight: 400;\"> instead of the sum. The mean (or arithmetic mean) is the sum divided by how many numbers there are:<\/span><\/p><p><span style=\"font-weight: 400;\">Mean = 100 \u00f7 10 = <\/span><b>10<\/b><\/p><p><span style=\"font-weight: 400;\">There\u2019s a shortcut here as well. The sum of the first n odd numbers is n\u00b2, so dividing by n leaves just <\/span><b>n<\/b><span style=\"font-weight: 400;\">. The average of the first ten odd numbers is 10, the average of the first 25 is 25, and so on.<\/span><\/p><p><span style=\"font-weight: 400;\">This is why multiple-choice questions like \u201cThe A.M. of the first ten odd numbers is (1) 10 (2) 100 (3) 1000 (4) 1\u201d have <\/span><b>10<\/b><span style=\"font-weight: 400;\"> as the answer. Option 2 (100) is the sum, which is the most common wrong pick.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-5d7e6db\" data-id=\"5d7e6db\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-49cd008 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"49cd008\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-84334d4\" data-id=\"84334d4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-ee0b672\" data-id=\"ee0b672\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cbb27a8 elementor-widget elementor-widget-heading\" data-id=\"cbb27a8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Common Mistakes to Avoid<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7273ecb elementor-widget elementor-widget-text-editor\" data-id=\"7273ecb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Having seen how students approach this question, these are the slip-ups that come up most:<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Mixing up sum and mean.<\/b><span style=\"font-weight: 400;\"> The sum is 100, and the mean is 10. Read the question twice.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Starting from 0 or 2.<\/b><span style=\"font-weight: 400;\"> Zero is even, and 2 is even. The list begins at 1.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Stopping at 17 or running to 21.<\/b><span style=\"font-weight: 400;\"> Count your terms. There should be exactly ten, ending at 19.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Using the even number formula by mistake.<\/b><span style=\"font-weight: 400;\"> The sum of the first n <\/span><i><span style=\"font-weight: 400;\">even<\/span><\/i><span style=\"font-weight: 400;\"> numbers is n(n + 1), which for ten numbers is 110. The odd-number sum is n\u00b2. Keep the two straight.<\/span><\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3e29286\" data-id=\"3e29286\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c17f6c9 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"c17f6c9\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-124287b\" data-id=\"124287b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1dd197f\" data-id=\"1dd197f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7070155 elementor-widget elementor-widget-heading\" data-id=\"7070155\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Why Does This Pattern Matter?<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-53ad069 elementor-widget elementor-widget-text-editor\" data-id=\"53ad069\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This isn\u2019t only a classroom trick. The \u201codd numbers build squares\u201d idea shows up in:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Algebra and number patterns<\/b><span style=\"font-weight: 400;\">, where it\u2019s often the first proof by pattern students meet<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Mathematical induction<\/b><span style=\"font-weight: 400;\">, where \u201csum of the first n odd numbers = n\u00b2\u201d is a classic worked example<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Mental arithmetic<\/b><span style=\"font-weight: 400;\">, where it lets you check large sums quickly<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Computer science<\/b><span style=\"font-weight: 400;\">, where programmers use the same closed-form idea to replace slow loops with a single calculation<\/span><\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-d28d4b8\" data-id=\"d28d4b8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bc02221 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"bc02221\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-6c61c41\" data-id=\"6c61c41\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-bdef610\" data-id=\"bdef610\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3ce2279 elementor-widget elementor-widget-heading\" data-id=\"3ce2279\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Practice Questions<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-044d243 elementor-widget elementor-widget-text-editor\" data-id=\"044d243\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Try these before checking the answers.<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">What is the sum of the first 15 odd numbers?<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">What is the average of the first 12 odd numbers?<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">What is the sum of the odd numbers from 11 to 29? (Hint: subtract the sum of the first five odd numbers from the sum of the first fifteen.)<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">How many odd numbers must you add together to reach a total of 144?<\/span><\/li><\/ol><p><b>Answers:<\/b><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">15\u00b2 = 225<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">12<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">15\u00b2 \u2212 5\u00b2 = 225 \u2212 25 = 200<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">12, because 12\u00b2 = 144<\/span><\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-fa864d8\" data-id=\"fa864d8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6bc3e73 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"6bc3e73\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-09e3550\" data-id=\"09e3550\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-0144845\" data-id=\"0144845\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-77940d3 elementor-widget elementor-widget-heading\" data-id=\"77940d3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Conclusion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-460220d elementor-widget elementor-widget-text-editor\" data-id=\"460220d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The sum of the first ten odd numbers is <\/span><b>100<\/b><span style=\"font-weight: 400;\">. You can reach it by pairing numbers to 20, by squaring 10, or by building a 10 \u00d7 10 square one L-shaped layer at a time. All three methods give the same answer, and that agreement is what makes the pattern so satisfying.<\/span><\/p><p><span style=\"font-weight: 400;\">Next time you see a question like this, try the visual method at least once. It\u2019s the one people tend to remember long after the formula fades.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-33a3a22\" data-id=\"33a3a22\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9eb3d27 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"9eb3d27\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c00bb7e\" data-id=\"c00bb7e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-619651e\" data-id=\"619651e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-acbf18f elementor-widget elementor-widget-heading\" data-id=\"acbf18f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Frequently Asked Questions\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-416173e elementor-widget elementor-widget-accordion\" data-id=\"416173e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"accordion.default\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-6851\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-6851\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">What is the sum of the first ten odd numbers?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-6851\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-6851\"><p><span style=\"font-weight: 400;\">\u00a0It\u2019s 100. The numbers 1, 3, 5, 7, 9, 11, 13, 15, 17 and 19 add up to 10\u00b2, which is 100.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-6852\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-6852\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">What is the formula for the sum of the first n odd numbers?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-6852\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-6852\"><p><span style=\"font-weight: 400;\">The sum equals n\u00b2. For ten numbers, that\u2019s 10\u00b2 = 100.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-6853\" class=\"elementor-tab-title\" data-tab=\"3\" role=\"button\" aria-controls=\"elementor-tab-content-6853\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">What is the 10th odd number?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-6853\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"3\" role=\"region\" aria-labelledby=\"elementor-tab-title-6853\"><p><span style=\"font-weight: 400;\">It\u2019s 19. Use the rule 2n \u2212 1 with n = 10.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-6854\" class=\"elementor-tab-title\" data-tab=\"4\" role=\"button\" aria-controls=\"elementor-tab-content-6854\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">What is the average of the first ten odd numbers?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-6854\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"4\" role=\"region\" aria-labelledby=\"elementor-tab-title-6854\"><p><span style=\"font-weight: 400;\">\u00a0It\u2019s 10. Divide the sum, 100, by the ten numbers.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-6855\" class=\"elementor-tab-title\" data-tab=\"5\" role=\"button\" aria-controls=\"elementor-tab-content-6855\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Is the sum of the first ten odd numbers odd or even?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-6855\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"5\" role=\"region\" aria-labelledby=\"elementor-tab-title-6855\"><p><span style=\"font-weight: 400;\">It\u2019s even. Ten odd numbers added together always give an even total, because pairs of odd numbers add up to even numbers.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"What is the sum of the first ten odd numbers?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">\\u00a0It\\u2019s 100. The numbers 1, 3, 5, 7, 9, 11, 13, 15, 17 and 19 add up to 10\\u00b2, which is 100.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"What is the formula for the sum of the first n odd numbers?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">The sum equals n\\u00b2. For ten numbers, that\\u2019s 10\\u00b2 = 100.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"What is the 10th odd number?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">It\\u2019s 19. Use the rule 2n \\u2212 1 with n = 10.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"What is the average of the first ten odd numbers?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">\\u00a0It\\u2019s 10. Divide the sum, 100, by the ten numbers.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"Is the sum of the first ten odd numbers odd or even?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">It\\u2019s even. Ten odd numbers added together always give an even total, because pairs of odd numbers add up to even numbers.<\\\/span><\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1d844c3\" data-id=\"1d844c3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Sum of the First 10 Odd Numbers: 3 Easy Methods with Visual Proof The sum of the first ten odd [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[1],"tags":[],"class_list":["post-28333","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - 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